For a positive small parameter $ϵ$, an order-one mismatch between the reduced solution $y_0=f(x)$ and a prescribed right endpoint value $y(1)=b$ creates a right boundary layer. Balancing $ϵy^{\prime \prime }$ against the order-one reaction term gives width $\sqrt{ϵ}$. In the inner variable $X=(1-x)/\sqrt{ϵ}$, the leading equation is $Y_0^{\prime \prime }=Y_0-f(1)$. Its boundary and matching conditions are $Y_0(0)=b$ and $Y_0\to f(1)$ as $X\to \infty$, soThe leading additive composite expansion is $f(x)+(b-f(1))e^{-(1-x)/\sqrt{ϵ}}$, obtained by subtracting the common overlap value. For the nonlinear perturbation $ϵy^{\prime \prime }=y-x^2+ϵ(2+y^2)$ with zero endpoint values, the first two outer expansion terms are $x^2-ϵx^4$, and the leading right-layer term is $1-e^{-X}$.

For $ϵy^{\prime \prime }+x^{\alpha }{y}^{\prime }+y=0$ near zero, dominant balance gives a diffusion-drift layer of thickness ${ϵ}^{1/(\alpha +1)}$ for $-1<\alpha <1$, with the reaction term negligible. At $\alpha =1$, all three terms survive on the scale $\sqrt{ϵ}$. For $\alpha >1$, diffusion and reaction first balance on wavelength $\sqrt{ϵ}$, while the drift damps the oscillation on the larger envelope scale ${ϵ}^{1/(\alpha +1)}$. Removing the first derivative gives the envelope factor $\exp [-x^{\alpha +1}/(2ϵ(\alpha +1))]$. For $\alpha \le -1$, no shrinking algebraic diffusion layer of this regular bounded type is obtained.

Split the integral at $ϵ\ll \delta \ll 1$. Replacing the numerator by one below the cutoff gives ${ϵ}^{-1}\arctan (\delta /ϵ)$, with an error bounded by $\frac{1}{2}\log (1+{\delta }^2/{ϵ}^2)$. Above the cutoff, replacing the denominator by $x^2$ has relative error $O({ϵ}^2/{\delta }^2)$, and integration by parts gives $1/\delta +\log \delta +O(1)$. The inner $-1/\delta$ cancels the outer $1/\delta$, and all errors are $o(1/ϵ)$, proving the leading constant independently of the intermediate cutoff.

The conditional expectation of $\sin B_T$ at time $t$ is $e^{-(T-t)/2}\sin B_t$, by independent Brownian increment. Itô's formula cancels its time derivative against its second spatial derivative and leaves the displayed stochastic integral. The integrand is bounded, so Itô isometry verifies its square integrability.

A twin plane is a compatible planar interface between differently oriented crystal variants. In a piecewise homogeneous elastic deformation its reference normal is the $N$ in a rank-one connection between the two variant deformation gradients. Continuity of the deformation holds along this plane even though its derivative changes. Other material planes crossing the interface can kink after deformation.

A martensitic variant is one of the lattice deformations corresponding to the low-temperature crystal phase. The parent crystal's discrete symmetries generate different variants. In a nonlinear elasticity model each representative $U_i$ supplies a hyperelastic energy well $SO(3)U_i$. A representative need not itself be a symmetric stretch tensor; multiplication by a proper rotation leaves the same well.

A function of the current state of a Markov chain can discard information needed to predict its next value. A sufficient condition preserving the Markov property is strong lumpability: all original states with the same recorded value must give the same transition probabilities to every recorded class.

For the two-column model in growth rate of two-column sticky deposition, the pair of column heights is Markov, but its maximum alone is not. Starting empty, the maximum-height histories $(1,1,2)$ and $(1,2,2)$ both end at height two. The former leaves unequal columns, so the next height increases with probability $1/2$; the latter leaves equal columns, so it increases with probability one. Each history has positive probability. Their different predictions at the same time and current value violate the Markov property. By contrast, the absolute height difference retains exactly the information needed for its own transitions.

A total function computed by a halting logspace transducer. Its output may be much longer than its work tapes. Logspace-computable functions compose using logspace composition by recomputation.

For integers $b>1$ and $n>0$, the integer logarithm is the largest nonnegative integer $i$ such that $b^i\le n$. Equivalently,The equivalence follows from the monotonicity of exponentiation in its exponent and the definition of the floor function. For example, $3^4=81\le 100<243=3^5$, so the base-three integer logarithm of $100$ is $4$.

A multiplication loop maintains $p=b^i\le n$ and accepts another factor $b$ while $pb\le n$. Repeated squaring makes the search faster: precompute $b^{2^k}$ starting at $k=0$, and consider these factors in decreasing $k$. Include a factor when the accumulated product remains at most $n$. If $L=\lfloor {\log }_{b}n\rfloor$, the loop invariant before considering $k$ is $i\le L<i+2^{k+1}$. Accepting $2^k$ exactly when $i+2^k\le L$ halves that uncertainty, proving the greedy algorithm returns $L$. This takes $O(1+\log (1+{\log }_{b}n))$ arithmetic operations; it is not the same claim as this bound on bit complexity.

For positive operands in bounded integer arithmetic, test $p\le n\mathrm{DIV}q$ before multiplying by $q$. Integer division makes this equivalent to $pq\le n$ while preventing integer overflow in a rejected candidate.

For $F(x)=1-[1+(x/\alpha {)}^{\beta -1}{]}^{-1}$, plotting the logits of $p_i=i/(n+1)$ against sorted log observations gives an approximately straight line under the model. The probability integral transform makes $F(X_{(i)})$ a uniform order statistic with expectation $i/(n+1)$, motivating these plotting positions. With those axes the slope is $\beta -1$ and the intercept is $-(\beta -1)\log \alpha$.

A bulb-volume change shifts the indicated temperature. Contraction raises the liquid column at a fixed temperature; expansion lowers it. Permanent and temporary changes require distinction when interpreting an ice-point check.^[ref]

The norm of a linear-system residual, used to measure how closely an approximate solution satisfies its linear system. It differs from the solution-error norm $\Vert x_{\ast }-x\Vert$. Their relationship depends on the matrix inverse and the condition number, so a small residual alone is not a scale-free relative-error guarantee.

At symmetric design points, the columns $1$, $x$ and $x^2-\overline{x^2}$ are mutually orthogonal. Their squared lengths give the diagonal entries of the normal-equation matrix and hence separate least-squares coefficient variances. At the seven integer points from minus three to three, these squared lengths are $7,28,84$.

Fit a quadratic mean at three temperatures $x_0,x_0\pm 1$, with equal replication $n$ at the off-center temperatures and any positive replication $m$ at the center. The three parameters fit the three group means exactly, giving the displayed estimates. Independent Gaussian errors make these estimates jointly normal. Their covariance is $\frac{1}{4}[\mathrm{Var}({\bar{Y}}_{+})-\mathrm{Var}({\bar{Y}}_{-})]=0$, so they are independent. Equal off-center replication is essential to this cancellation.

An ideal linear polarizer transmits the fraction ${\cos }^2\theta$ of incident linearly polarized light when its axis makes angle $\theta$ to the polarization. For a single photon, the corresponding conditional transmission probability is $|⟨\theta |0⟩{|}^2={\cos }^2\theta$, by the Born rule; rejection has probability ${\sin }^2\theta$. The classical field calculation projects its electric-field amplitude onto the transmission direction and squares that amplitude to obtain intensity. This is a conditional probability given the incident polarization and is not automatically the joint probability of passing one analyzer and failing another.

An operator on a complex Hilbert space is accretive when the displayed inequality holds throughout its domain. Its negative is a dissipative operator. For a bounded everywhere-defined operator, expansion gives $\Vert (I+T)x{\Vert }^2\ge \Vert x{\Vert }^2$. Its adjoint operator satisfies the same accretivity condition. Consequently $I+T$ has closed range and trivial kernel, and its range is dense because its adjoint has trivial kernel. Thus $I+T$ is onto and $\Vert (I+T{)}^{-1}\Vert \le 1$. Boundedness and the full domain are material here; for unbounded operators accretivity alone need not imply surjectivity.

A linear interpolant is the unique polynomial of polynomial degree at most one taking prescribed values $y_a,y_b$ at distinct interpolation nodes $a,b$. It is an affine function, including the constant function case. For $a\le x\le b$, its two displayed coefficients are nonnegative and sum to one, so changing each sampled value by at most $\delta$ changes every interpolated value by at most $\delta$. Differentiation instead uses the difference of the two samples divided by $b-a$, causing noise amplification by differentiation of a linear interpolant.